To solve this problem:

1. Analyze the problem setup:

   • The lines $k$ and $n$ are parallel ($k \parallel n$).
   • The transversal $m$ intersects both parallel lines.
   • There are two given angles:
     • $(6y - 60)^\circ$ and $z^\circ$ are consecutive interior angles.
     • $60^\circ$ is an alternate interior angle with $z^\circ$.

2. Use angle relationships:

   • Consecutive interior angles: The sum of consecutive interior angles on the same side of the transversal is $180^\circ$. Therefore: $(6y - 60) + z = 180$
   • Alternate interior angles: $z$ and $60^\circ$ are alternate interior angles, so they are equal: $z = 60$

3. Solve for $z$: From the alternate interior angle property: $z = 60^\circ$

4. Solve for $y$: Substitute $z = 60$ into the first equation: $(6y - 60) + 60 = 180$ Simplify: $6y = 180$ Divide by $6$: $y = 30$

Final Values:

$z = 60, \quad y = 30$

Let me know if you would like additional explanation or related insights!

Related Questions:

1. What are the other angle relationships formed by a transversal cutting parallel lines?
2. How can you identify alternate interior angles versus corresponding angles?
3. If $z$ is given, how can we deduce the other angles in the diagram?
4. Why is it significant that $k \parallel n$ in this scenario?
5. What would happen if $k$ and $n$ were not parallel?

Tip: Always check if lines are parallel before applying these angle properties!